F70CF Continuous-Time Finance
Lecturer: Mark P. Owen
Aims
This module develops the theory and practice of financial derivatives pricing in continuous time. The module contributes towards exemption from the CT8 examination of the Faculty and Institute of Actuaries
Summary
We shall introduce the Black Scholes model, and look at some simple extensions to treat dividends/currencies.
We will look at bond pricing and the term structure of interest rates in an arbitrage-free framework. We will also cover how to price bond derivatives, and look at models of credit risk.
This module will concentrate on the application of stochastic modelling, however it will also require some of the basics of the theory of stochastic process, in particular a knowledge of Brownian motion and its properties.
Learning outcomes
At the end of studying this module, students should be able
to:
- Demonstrate a knowledge of Brownian motion and its properties;
- Show how to calibrate the Binomial model as an approximation to Brownian motion using empirical data;
- Apply Ito's Formula, the Girsanov theorem and the martingale representation theorem;
- Work with martingale measures, and understand their connection with arbitrage free/complete markets;
- Understand the concepts of replication, hedging, and delta hedging in continuous time;
- Derive the Black-Scholes formula and the Black-Scholes PDE;
- Price contingent claims (in particular European style options and forward contracts);
- Extend the Black-Scholes formula to foreign currencies and dividend paying stocks;
- Understand the role of the Greeks in portfolio risk management;
- Derive relationships between forward interest rates, spot rates and zero coupon bond prices;
- Understand issues involved in selecting and using short rate models for pricing bonds and bond derivatives;
- Manipulate explicit bond price formulae for the Vasicek and CIR models, and derive the implied forward rate curves;
- Define the different approaches to modelling credit risk;
- Define and apply the Merton model for credit risk to price simple corporate bonds and calculate credit spreads;
- Define and apply the 2-state model for credit risk with deterministic and stochastic transition intensities.
Reading
The following books are recommended:
- Hull, J., (2000) Options, futures and other derivative
securities, 4th ed, Prentice Hall;
- Baxter, M. and Rennie, A. (1996), Financial
calculus, Cambridge University Press;
Students may also find the following books useful:
- Luenberger, D.G. (1998) Investment science. Oxford
University Press.
- Bingham, N. H. & Kiesel, R. (1998) Risk neutral
valuation. Pricing and hedging of financial derivatives.
Springer Verlag;
- Björk, T. (1998) Arbitrage theory in continuous
time. Oxford University Press;
Assessment
2 hour end-of-year examination (100%).
Help
If you have any problems or questions regarding the module, you are encouraged to contact the lecturer.
Module web page
Students who have already registered for this module can find further information and learning materials within the Univerity's Online Learning Environment, Vision.
Syllabus
- Theory of Martingales in continuous time
- Brownian motion; definitions and properties
- Brownian motion as the limit of a binomial random-walk process
- Introduction to stochastic integration, stochastic differential equations and Ito’s formula
- Geometric Brownian motion; the Ornstein-Uhlenbeck process
- Introduction to Girsanov's theorem and the martingale representation theorem
- The Black-Scholes model
- Derivatives pricing using the Black-Scholes model using the martingale and PDE approaches to pricing
- Extensions to foreign currencies and dividend-paying stocks
- Portfolio risk management using the Greeks
- Introduction to interest rate models
- Introduction to credit risk models
